\(\lim _{\mathrm{x} \rightarrow(\pi / 2)}[\\{\sin (\cos \mathrm{x}) \cos \mathrm{x}\\} /\\{\sin \mathrm{x}-\operatorname{cosec} \mathrm{x}\\}]=?\) (a) 0 (b) 1 (c) Limit does not exist (d) \(-1\)

Trigonometric identities are mathematical equations that express one trigonometric function in terms of another. These identities are incredibly useful when it comes to simplifying expressions and solving problems involving trigonometric functions.

For example, in our exercise, the cosecant function is defined as \(\operatorname{cosec} x = \frac{1}{\sin x}\). Using this identity, we can rewrite complex trigonometric expressions into simpler forms that are more manageable. By understanding and employing these identities, students can break down and evaluate limits of functions that, at first glance, might seem intimidating.

Applying trigonometric identities is crucial when approaching limits involving trigonometric functions, as they often allow for the simplification needed to determine the behavior of the function as it approaches a certain value. When we use identities like \(\cos^2(x) + \sin^2(x) = 1\) or \(1 + \cot^2(x) = \csc^2(x)\), we're essentially 'translating' trigonometric expressions to reveal their underlying characteristics.

Indeterminate forms occur when evaluating a limit yields an expression that is not immediately clear or is undefined, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These forms need additional manipulation to resolve.

In the provided exercise, after simplifying the expression and plugging in the limit values, we end up with the indeterminate form \(\frac{0}{0}\). It's important to note that an indeterminate form does not mean the limit does not exist; however, it signifies that we can't determine the limit's value from the information given in that form alone.

To resolve indeterminate forms, one can use various techniques such as L'Hôpital's rule, algebraic simplification, or trigonometric identities—like the ones in our exercise—to transform the indeterminate expression into a determinate form where the limit can be directly evaluated.

Evaluating limits is a fundamental aspect of calculus that involves finding the value that a function approaches as the input approaches some value. Limits can often be straightforward, but they can also present challenges, such as with indeterminate forms or functions that oscillate near the point of interest.

In our exercise, we attempted to evaluate the limit of a trigonometric function as \(x\) approached \(\frac{\pi}{2}\). Each individual term of the function is evaluated at the limit, and any continuous functions directly allow for the input value to be substituted. However, when we encounter an indeterminate form such as \(\frac{0}{0}\), this standard approach does not yield a clear result, suggesting that further analysis or methods should be employed.

Understanding the behavior of trigonometric functions near certain key points, and knowing how to manipulate and simplify expressions using trigonometric identities are integral to successfully evaluating limits. When faced with more complicated limits, additional techniques like factoring, conjugation, or L'Hôpital's rule might be necessary to find a solution.